# Energy Wells & Harmonic Oscillators

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### How to Find a Wave-Function Equation in an Infinite Square Well

Infinite square well, in which the walls go to infinity, is a favorite problem in quantum physics. To solve for the wave function of a particle trapped in an infinite square well, you can simply solve

### How to Find the Wave Function of the Ground State of a Quantum Oscillator

In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.

### How to Use Creation and Annihilation Operators to Solve Harmonic Oscillator Problems

Creation and annihilation may sound like big make-or-break-the-universe kinds of ideas, but they play a starring role in the quantum world when you're working with harmonic oscillators. You use the creation

### How to Shift the Symmetric Square Well Around the Origin

Infinite square wells, in which the walls go to infinity, are a favorite in quantum physics problems. In some instances, you may want to shift things so that the square well is symmetric around the origin

### How Particles Pass Through Potential Barriers That Have Less Energy

When you are working with potential barrier of height V0 and width a where E > V0, this means that the particle has enough energy to pass through the potential barrier and end up in the

### How to Find the Energy Eigenstate of a Harmonic Oscillator in Position Space

In quantum physics, you can use operators to determine the energy eigenstate of a harmonic oscillator in position space. The charm of using the operators

### How to Find Any Excited State of a Harmonic Oscillator

If you can determine the wave function for the ground state of a quantum mechanical harmonic oscillator, then you can find any excited state of that harmonic oscillator.

### How to Find the Energy Level of a Harmonic Oscillator: An Example

Your quantum physics instructor may ask you to find the energy level of a harmonic oscillator. The best way to learn how is through an example. Say that you have a proton undergoing harmonic oscillation

There are plenty of free particles — particles outside any square well —in the universe, and quantum physics has something to say about them. The discussion starts with the Schrödinger equation:

### How to Estimate a Particle's Location by Applying Schrödinger's Equation to a Wave Packet

If you have a number of solutions to the Schrödinger equation, any linear combination of those solutions is also a solution. So that’s the key to getting a physical particle: You add various wave functions

### How Particles Tunnel Through Potential Barriers That Have Greater Energy

When a particle doesn't have as much energy as the potential of a barrier, you can use the Schrödinger equation to find the probability that the particle will tunnel through the barrier's potential. You

### Measuring the Energy of Bound and Unbound Particles

In quantum physics, you can solve for the allowable energy states of a particle, whether it is bound, or trapped, in a potential well or is unbound, having the energy to escape.

### How to Find Square-Well Energy Levels

In quantum physics, if you know the boundary conditions of a square well, you can find theenergy levels of an electron.

The equation

### How to Find the Normalized Wave Function for a Particle in an Infinite Square Well

In quantum physics, if you are given the wave equation for a particle in an infinite square well, you may be asked to normalize the wave function. For example, start with the following wave equation:

### Find the Wave Function of a Particle in an Infinite Square Well over Time

In quantum physics, you can use the Schrödinger equation to see how the wave function for a particle in an infinite square well evolves with time. The Schrödinger equation looks like this:

### How to Find the Eigenfunctions of L2 in Spherical Coordinates

Your quantum physics instructor may ask you to find the eigenfunctions of L2 in spherical coordinates. To do this, you start with the eigenfunction of

### How to Determine Harmonic Oscillator Eigenstates of a System

In quantum physics, when you have the eigenstates of a system, you can determine the allowable states of the system and the relative probability that the system will be in any of those states.